 Nice. Good morning, good morning, everyone. Hope you're doing well. Welcome to another live stream. Today, today, what are we in? We're December 14th, 2020. And this is an anniversary live stream because three years ago, as of today, we started live streaming on Twitch. So I was reminded that today was our anniversary stream and I ended up sort of setting up this live stream to do especially sort of focus on mathematics since everything's layered on mathematics. So we decided to do a math live stream for our anniversary stream, third year anniversary live streaming on Twitch, which is fantastic. Thank you, Elder God, for reminding us because I wouldn't have remembered, I wouldn't have checked it. Good evening from Holland. Good evening, Holland. How are you doing? Good morning from my part. Kebabs, how are you doing? Happy anniversary, bro. Thank you very much. Three years, wow. I looked at the first stream that we did where I actually took it in and edited it. I did a little intro at the beginning. So it was pretty cool, actually. Gravity of the situation. How are you doing? Hope you're doing well. Doing well, brother. Thank you very much. Definitely wanted to get a stream in today. And I'm glad we are. I'm glad we are. Lonely Piggy, how are you doing? Good afternoon to you and Chad. And congratulations on three years on Twitch. What a ride. What a ride. What a ride. What a fantastic ride, really. We're up to, what are we in with subs right now on Twitch? We're up to 4,200 people that are following us on Twitch, which is fantastic. We're up to 33,000 on YouTube. Our bitchip numbers are still pretty low, but that's okay. That'll slowly pick up and we're going to open up new platforms. We're now sharing stuff on SoundCloud as well. We got Discord going, which is amazing. Our Discord channel is fantastic. It's basically reduced a lot of places I was going to get information. And I'm coming to our Discord page to get information and share information. So that's fantastic. We need an only fans. Maybe. Maybe. I want to keep it open, right? But for sure, at some point we'll have to as the numbers increase. But right now, I think the conversation is really sweet because we have a lot of people that have been here a long time discussing things and then we get new people coming in. They join the conversation, right? So I do want to make sure we're open for people to join us because for me it is about sharing as much information as possible, right? Really. It really is. As a technocrats and centralized powers and stuff like this, even universities, schools, censor information, basically do book burnings. We need to open up the platforms and just go crazy with it, right? So, you know, it is what it is. And Kebabs, I appreciate the free content of my pleasure, man. I consume free content. I share free content, right? I try not to be a hypocrite on the deal, right? So if I'm consuming a lot of work from a lot of different people, I will also share as much as I can. And I think I've done a pretty good job of it and we've done a pretty good job of it, right? Aside from that, gang, we're doing a math stream. I'm going to do my little intro as always if you want to know what this is all about. I am on patreon.patreon.com forward slash chico, C-H-Y-C-H-O. Everything is layered on mathematics. I don't put anything behind paywalls, share and share alike. If you want to know what this is about, on this side, Patreon is a great way to do so. And if you want to support this work, Patreon is a great way to support this project. And for those of you who've been supporting this project, thank you very much for the support gang. I appreciate it. My man, always love when you are on awesome, awesome Lark Park. Three years, three years. Rock and roll. It's math time, folks. It's math time, folks. We are live streaming on Twitch, twitch.tv forward slash chico live, C-H-Y-C-H-O-L-I-V-E. If you want to participate in the chat with a little happy face going on there, Twitch is where you want to be, guys. Elder Gudak. Wait a second. Elder God is changing your name. God UK. Elder... I read it as Elder Gudak, but it's Elder God UK. Nice. Elder Gudak. How funny. How come the name changed, brother? Had my college math final got 91%. Awesome slick mic. That's the way to go. Above 90 means you're rocking it, right? It means you're rocking it. Gang, for those of you who've been here on Twitch, supporting this project on Twitch, subscribing, following, bits, chatting it up, showing up on live stream, sharing information, being on our Discord, thank you very much for the support. I wonder where Elder God UK is from. Third year anniversary. Oh, third year anniversary. You changed it up. Celebration, celebration, man. Celebration, celebration. I do announce these live streams 30 minutes before we go live on Parler, Elo, Mines, VK, Gav, and Twitter. And we do share additional content there. And anytime you want, you can come to our Discord on our Twitch page and type in exclamation mark social in our chat and you'll get all the social media links popping up, as well as you can see over here, where is it? Before it disappears, right there. There's our Discord page as well, where there's a lot of people sharing a lot of information, which is fantastic. Okay, so you're welcome to join us on all those platforms. I do upload audio where we don't have any visuals, on to SoundCloud.com. And they should be available on your favorite podcasting platform. And this is Mathematics. And we're going to upload this both to BitShoot and YouTube. And if you're on those platforms, you can support this work by subscribing, turning on notifications, guaranteed to go notifications on BitShoot, maybe on YouTube. Some people are saying they're beginning it. Some people are saying they're not getting it. So maybe YouTube sends it out for certain things, not other things. I don't know what's going on with YouTube. Serious glitches there. Both intentional, unforeseen, a lot of censorship going on. So I think they're just rolling out algorithms that aren't tried and tested. And they're trying to eliminate a lot of discussion on YouTube. Hopefully we're not one of them. And if we are, if we see us disappearing, hopefully not BitShoot. And at the beginning of 2021, we're going to introduce at least a third video sharing platform. And once we introduce the third video sharing platform, the odds are, I'm going to start uploading the previous math content, specifically language of mathematics, math and real life, and a lot of the ASMR mathematics we've had previously onto those platforms as well. So there's going to be a wave of videos coming on. Gina, how are you doing? Hope you're doing well. And welcome to another live stream. Felix, how's life? Hope you're good, brother. Hope you're good. My new YouTube account just got a video removed after two hours. Dude, elder god, I don't think by the time I subscribe to your channel, I think you're going to get knocked out. Nate, how are you doing? Hello, hello. Sean, yo, how is life? Welcome, welcome. Let me take these guys down. Thawing, thawing, thawing, thawing, thawing. Man, this is math, but it is an anniversary stream. And lonely piggy, what do I got? What do I got? You didn't ask. Apples and peanut butter. Yum. Apples and peanut butter. You got your carbs with the apples and fiber with the apples. You got protein with the peanut butter. Pretty good. Fantastic snack. Glorious snack. Thanks, lonely piggy. Felix says, I had an idea for a stream yesterday that I posted during the comic reading. I wasn't there for the whole stream, so I don't know if you caught it. I don't know if I caught it. What was it, Felix? What do you got? They didn't like my bank robbing clown. Oh, the bank robbers. The one you sent me. That was from Brazil or something with the clowns and they were wrapping it up with the guns in the car. That was crazy, man. Like, they're just having a party going for a bank robbery. It's like, damn. Was it in Brazil? Yeah. They didn't like it. I mean, there was no violence shown or anything, so I don't know why they would take it down. They got stuff on there. That's crazy, YouTube. Legendary Rob Boss. What are we doing? And the fat with the peanut butter. Yeah, and the fat with the peanut butter. Peanuts aren't really nuts. They're legumes. They're beans, right? Always snacking. Always snacking, John Yope. Reading points. 100 messages. Johnny. Johnny X. Johnny. How are you doing? Hi, homie. Hope you're having a good day. Love from Texas. Ah, lots of love right back, Johnny. Thank you very much. Appreciate it. Appreciate it. Best Ghost of Canada sends you all the love. Felix, I was thinking we could run a stream where you play some 10, 15 minute online chess games with some of the people in chat. I don't know if it's feasible, but I thought it was a fun idea. That would be a fun idea. I've been thinking about doing it for backgammon, right? Set up the thing, roll the dice, and then we do it with backgammon. And we haven't done a backgammon stream for ever. And I want to get back into chess. I pulled out my chess board because my partner, after watching Queen's Gambit, I think Queen's Gambit, the TV show, and the last episode is not worth it, but up to the last episode it was really good. It was well done. It was a fun show. I sort of added the bling-bling to chess. Not that you really need to add bling-bling and CG and stuff like this, but it's sort of an introduction, sort of something to get people hyped up for chess, right? Hopefully people aren't laying down in bed at night trying to get 3D chess happening on their ceilings. But I brought up my chess board to teach her how to play chess. So once I get back into it, I might. I miss playing chess, actually. I feel like Ground Ward and some of the mods were up for it, I believe. Cool. Yeah, chess is amazing. At the time when I was playing, I played a lot when I was a kid. I started playing chess, I think. And gang, if you have kids, if you have cousins, if you have nephews and nieces, if you have friends that have kids, that you're looking to help in their education and their mental development and their emotional development, try to empower them with the ability for them to be able to do critical thinking and connect the dots and become more intelligent. Start playing board games with them. Two of the board games I would say you should start playing with them is one is chess, the other one is backgammon. Chess and backgammon. Those are the two board games you should try to start playing with kids at a really, really young age. Like four years old. Four years old. I was probably playing backgammon in chess when I was like five. It's a great time. Kids can pick it up. Speed of Anzalastah. There are other board games as well. One of the other ones which is brilliant is Mastermind. When you line up the colors and the other person has to guess which colors. That one might require for them to be a little bit older, but it should work with five or six as well. Slick Mac. I remember watching you play backgammon with your mother, with my grandma or grandma. It was awesome. It was. Oh brother, Slick Mac. I was trying to get her over here before this whole pandemic stupid thing hit to try to do more live streams with her. It just didn't happen. So we're in different cities right now and as soon as I can get her over here she's getting pretty old, man. She's getting pretty old, but I haven't seen her for a few months now. I was going there. I was seeing her on a weekly for a long time, bi-weekly. At least once a month I would go there and play backgammon for the weekend. And it's amazing too. And by the way, playing backgammon in chess is not just for kids, right? Like for me, one of the mental stimulations that my grandmother was getting because as you get older the body starts giving out, right? The only absolute in life has changed. So as people get older, one thing you need to do as you become senior citizen or what not you have to keep your mind engaged. Because if you don't keep your mind engaged you can get sick, you can get into dementia Alzheimer's and I know they're they're sort of connected and not connected but you need to stimulate the mind and that was my way of stimulating my grandmother's mind. She was pretty damn good at it, right? So show respect to the elders, man. Educate the kids. Spend time with your elders. Really. Lark bar chess is brilliant. We love to get into the habit of it. Yeah, indeed. Me too. When I was a kid, like daily. Okay. Emily, I've been playing a lot with my dad. We're gonna play when he is off work. You really should get back into it because it's so fun. Yeah, I know. I know. And gang. Free Assange. Free Assange. Free Assange, right? Emily, board games are so much fun for the brain. If you play a strategic game that's fun. Kids will build strategy skills young. Yeah. 100% Emily. I'm 100% with Emily on this one. System Veil. Can you prove that you can't have a set with five or more integers where none of their differences are divisible by four? Let me think about it. How would we go about this? Let me get caught up with chat. How's the chat going? Okay, I'm gonna keep this in mind. System Veil. Let me just try to get caught up with chat. I had a friend in H... H.S. H.S. who was taught chess. I can't be Hong Kong. H.S. who was taught by his extremely authoritarian father like drill sergeant stuff would get really mad and knock pieces over when I would start to win because I was self-taught. And that, as far as I'm concerned I don't want to say it's bad parenting but it's it's bad parenting. Really. If you're stressing out a kid to a level where they can't handle losing, it's... you're not teaching them right, right? Loss is part of life. Learn to deal with it young. Okay? Learn from it. Don't teach children not to throw ant-adults, not to throw tamper-tandrums. Oh, high school. H.S.S. high school. Okay, cool. I caught up on that over here. Jichou, have you ever heard of the 42 laws of mat, similar to commandments, but it comes from kemat, ancient Egypt. No. And they were our guidelines for religious living. No. I decided to record myself speaking to them as if I live up to them. And I'm going to listen first thing in the morning. Okay, why not? Meditation is a great thing. Felix, you're really lucky to still have your grandma I know brother around Jichou. And by one of my... All but one of my grandparents died before I was 10. Oh, no. And I was too young to appreciate my time with them, like I do with people now. Yeah, I feel you. I feel about that way about my great-grandmother. She passed away when I was like 19. I wish I'd spend more time with her. She loved me, but you always want to spend more, right? Slick Mac lost my grandad just last year to dementia. Oh, no. It was sad to see him go, but I kind of cope knowing it wasn't really him when he died. He fought dementia for three years. Okay. And three years is not that long fighting dementia. The people that I've known that their elders get dementia Alzheimer's or or other diseases, right? They find it a relief when they're gone, especially if they've been holding on for a long time. So take that you know a half way you wish, right? Yes. Free our hero, Sanj, and pardon another hero, Edward Stone, indeed. And support another hero, Chelsea Manning, indeed. Felix Jichou, could you explain how to use differentiation to find the turning point of a polynomial graph? Sure. We'll do that, Felix. But I've got to read up on the 42 one. The fact she's streaming on Twitch is absolutely incredible to me. It's surreal. Yeah, indeed. And in the system, there's two channels that I automatically host on my channel. One of them is Action for Sanj, which is a group of journalists and activists that have been getting together supporting Julian Sanj, sharing information. They're working their asses off, right? We started auto-hosting them when they were streaming on Twitch and Chelsea Manning. As soon as she hopped on Twitch to start streaming, I auto-hosted it as soon as I heard about it, right? Because I'm subscribed to her feed, like Twitter feed and stuff like this. So full support to Chelsea Manning, indeed. And I'm very happy to see her channel grow like mad. It's awesome. Emily, kids need to be taught losing isn't bad. We all lose sometimes, but we should push them to do the best they can, indeed, Emily. Ding Bobber, just as an extension, if these laws are upheld, your heart will become lighter than a feather and you will succeed in the afterlife. Okay. Emily, I lost my grandparents in the same week. Oh, I missed them so much, but it's always too late when you want to spend more time with them. Take time to spend with family while you can. Indeed, indeed. Greetings, blessings. So we're going to look at differentiation to find a turning point, but this other one, let me read this thing again. How is that? Can you prove that you can't have a set with five or more integers? Okay. Let me write this down. Five or more integers where none of their differences are divisible by four. None of their differences are divisible by four. So X, Y, Z, Q, and K. These are all integers and none of their differences are divisible by four. Oh man, that's so hard. I'm really brutal with these proof system veil. I really don't even know how to approach this, but basically the question is saying, can you prove that all the different combinations that you can make here subtracting them, which is going to be five choose two? Basically, you want to say this. X minus Y cannot equal cannot equal divisible by four. How would we say that? It's not even not equal to four. It cannot be divisible by four. So cannot be divisible by four. Cannot be divisible by four. How do we say that? So their difference. Oh, you would do this. You would say you would write it down. Divide it by four. Right? So if this was your function f of x and y, then it can't be divisible by four. So if you divide this, the remainder is not going to be zero. So how do you say the remainder is not going to... It blows me away. These proofs are hard for me, man. No. Just at least one difference is divisible. Not all of them. No. Just at least one is divisible. Not divisible or divisible. That none of their differences are divisible by four. Where none of their differences is divisible by four. That's what you wrote down first. But you're saying no. Just at least one difference is divisible by four or is not divisible by four. I wouldn't know how to approach it, brother. I really don't. Like I understand what you mean, but basically what you would do here is you would say this divided by this then the quotient plus it's going to have a quotient plus a remainder. Right? So for example, you're going to have this 27 divided by two. Right? Then if you divide this thing here, you do this. 27 goes once to one, bring the seven now. 6 subtract, you get one. Right? This is the Q. This is the divide dividend or whatever. This is the divisor the other way around. This is the remainder. So the remainder has to be a number where R is going to be anything but zero. Right? I wouldn't know how to do this. I don't know how to do this. I would recommend if you want to know this go to our discord page and post the question. There are people there that are doing pretty high level mathematics and they love this proof stuff. For me I've never been into the proofs. They twist me to a certain degree. I love the applications of mathematics. I like the algebra. I like data sets. The abstract stuff throws me off a little bit. It's just the way it is. I didn't go down that branch. I decided to change directions when I was doing geophysics. When I went to university, initially I was at one university where there was a lot of theoretical. We started doing proofs and the proofs during the exams, it would take like three pages to write down a proof of something. I was like, okay, I don't want to sit there and do three pages of proofs. That was the first year program. So I changed universities and I went to university. Apologies about that. As for the other question, how do we find the turning point for a function using differentiation? We're talking about calculus. Take a look at this. This is the way we do this. Let me make sure we're all doing good here. Yeah, we're all doing good here. Felix, differentiation to find the curve I believe is just dy over dx and then once you get your answer differentiate that. Once you get your answer set it equal to zero, that finds a turning point I believe. Watch this. Here, let's assume we have the following function. I'm just going to make it up. Let's make a let's make a third-degree function. Okay, to keep things simple. We're going to go through graph this thing and find all the critical points on it. Okay? So, f of x is equal to negative 3x squared plus 4x sorry, cubed plus 4x squared plus 5x plus 6. Okay? That's our third-degree polynomial. We're going to draw the graph here and slowly start graphing it. So, it's the second derivative, slick-mic. Teacher, did you teach any geometry math? Yeah. Yeah, I do teach geometry. I've been teaching math for about 20 years high school, mainly. Some university college, some elementary. So, let's see what we have in this right now. Okay? And we don't have the x-intercepts for this ch-ch-ch-ch-ch-ch. You know what? I'm going to switch this up. I'm just going to make it a... Okay, hold on a second, because we do need to ah, we could do synthetic division, but I really don't know well, this is going to have x-intercepts, but they're not going to be clean. Right? So, let me generate let me generate I'm just going to go to Wolf-Ram Alpha Wolf-Ram Alpha You could go to other sites as well, but I'd go to this one. This is the one I know. I'm just going to generate a quadratic or not a quadratic negative 3x plus 1 2x minus 3 and let's go x plus 1, 2. Let's see what this generates for us. And then I'm going to switch this up. Erase this guy. I'm going to give you a new quadratic. Right? I'm going to reverse-generate it. Hopefully it will give us a tweet. Yeah, there we go. Nice. This is our quadratic. Watch this. Hey, I said negative. I want it to be negative. It is negative. How come we made it positive? Oh, because there it is. Silly little bugger. Here's the quadratic function we got. Okay. Or not the quadratic. The cubic function we got. It's going to be negative 6x cubed minus x squared plus 19x minus 6. Okay? So we still ended up getting the negative 6, which is perfect. That's what we want. And gang, don't forget. Free Assange, Free Assange, Free Assange. Right? Felix, I'm slightly confused by it. But I thought that the second differential was to find whether the turning points were minimums, maximums, or a point of inflection. It should give you a point of inflection. Should it not? I'm probably wrong though. So I'll let you two explain. Let's go through it. Okay, let's go through it. Watch this. So let's put our first thing that we know here right now. F of 0 is your y-intercept. If you set x is equal to 0, because if this is f of x and this is your x, when x is equal to 0, that becomes 0 because 0 cubed. So that should be obvious. I hope to everyone. Second derivative is useful for finding the inflection points. Yeah, indeed. You use first derivatives to find the min-max. Yeah. And that's exactly what we're going to do. And thank you for flying QA. This becomes 0, this becomes 0. So our y-intercept is negative 6. So I'm going to go 1, 2, 3, 4, 5, 6. So 0 and negative 6. That's our y-intercept. Hopefully that comes out. Okay. That should be obvious. When x is 0, you set x0, this disappears, you get that. What we're going to do now is we do need the x-intercepts as well. So for the x-intercepts, what we're going to do, we're going to do synthetic division on this thing. Because the x-intercepts is one of the first things you want to find. Now what I'm going to do before we look for the x-intercepts, is I'm going to take the first derivative and the second derivative of this. Okay. So the first derivative of this is going to be, and it's a polynomial. So you take the 3 here and just for those of you that don't know how taking the derivatives of a polynomial works, if you have a polynomial f of x, let's say a x to the power of 5, or let's make this an n, doesn't make a difference. n plus whatever comes after it, it could be more terms that have x's in it. All you do, you take this kick it down so the derivative of this would be a times n and this would be n-1 plus whatever, and whatever is you do the same thing to any of the terms that have x to the power of integer, or you do the same thing to the rest of the terms. If one of the terms happens to be a number 3, then this just becomes 0 because the derivative of 3 is just 0 of an integer. I'm assuming you guys know this already. So what this becomes is the 3 comes down, multiplies the negative 6, this is negative 18. x to the power of 2, so you're basically taking out 1 degree from the polynomial. Minus 2 comes down that's just a 1 there, so it becomes 2 you kick that down, it becomes 1. This is a 1 comes down, multiplies 19 it becomes 19, this becomes x to the power of 0 because 1 minus 1 minus 1 is x to the power of 0, anything to the power of 0 is just 1, so it's 19 times 1. So that I'm going to erase this as well is your second derivative. Oh sorry, your first derivative your second derivative is going to be, take the derivative of the first derivative so this becomes negative 36x 2 comes down, multiplies 18 minus 2 and then that's it. Right? 19 just becomes 0, where 6 also became 0. Right? So what do we need to do? We need to find the x intercepts by setting f of x equal to 0 because when f of x is 0, that means your y is 0, so you're on the x axis. So if you set this is equal to 0, then you can factor this and find all your x intercepts. You're going to do the same thing for the second derivative and the sorry, the first derivative and the second derivative. Why do I keep on calling this the second derivative? The second derivative. Hopefully that shows up better. So let's do this first. Find x intercepts by setting f of x is equal to 0 and solving for x. Right? So set f of x equal to 0 and solve for x. Okay? So let's do this. I'm going to pull up my chair because we're going to have to do synthetic division. Right? So what we're going to do is negative 6x cubed minus x squared plus 19x minus 6 is equal to 0. Right? So what we need to do is find the values of x that make this equation equal to 0. Now the only way we can really do this is to factor this. Now this is a cubic. There's a formula you can use to factor cubic functions. I don't memorize it. I just do synthetic division. For synthetic division basically the factors that you can do manually, right? You don't need a computer or some kind of formula stuff to do this. Or possible factors of this divided by possible factors of this. So natural or let's call them rational factors. Right? So rational factors rational number factors are possible factors of negative 6 or 6 because the negative doesn't matter. It's plus or minus 1 plus or minus 2, plus or minus 3 and plus or minus 6 divided by possible factors of 6 which is the same thing again. Right? Plus or minus 1, plus or minus 2, plus or minus 3, plus or minus 6. So there's a whole ton of combinations here again. Right? Your possible factors could be 1, negative 1, 2, negative 2, 3, negative 3, 6, negative 6. They could also be 1 over 2, negative 1 over 2, 1 over 3, negative 1 over 3, 1 over 6, negative 1 over 6. They could also be 2 and the 1 doesn't matter because everything over 1 is just what I just read to you, right? It could be 2 over 2 which is just 1 so that's covered. It could be 2 over 3 or negative 2 over 3, 3 over 3, oh sorry and then 2 over 6, which is 1 over 3 so that's already covered and so on and so forth, right? Cruel joke, good afternoon Chihallix and welcome Chihominia. Good afternoon cruel joke. So what we're going to do is we're going to start off with the simple ones. I know that there's an integer factor of this because I set it up. The rest of them are fractions because as soon as we find a factor of this it means we took out an x from this degree and the next degree is a degree of 2 is a quadratic and we can just use the quadratic formula, right? So the way you do this is this you take the coefficients in front of the variables as well as the constant of the back, you write them out and this is basically long division, right? So you're going to go negative 6, negative 1, 19, negative 6 and what we're going to do is we're going to try out 1 and then negative 1 and then 2 and then negative 2 and then 3 and then negative 3. Now you don't have to do synthetic division for all of them, right? Starting off for 1 and negative 1 what you can do is use the remainder theorem, right? The remainder theorem says this for a specific value of x, if you sub it into the original function if the original function is equal to 0 then you're on the x-axis. Let me explain that to you using the y-intercept. So let's check this out. What was f of 0? f of 0 basically means you're going to set x is equal to 0 and solve for this thing, right? So it's going to be negative 6 times 0 cubed minus 0 squared plus 19 times 0 minus 6 so that's just 0, that's 0, that's 0. So this is equal to negative 6, so f of 0 is equal to negative 6 which is really your y-intercept, right? When x is 0, y is negative 6. Now this theorem is called the factor theorem or remainder theorem. This is called the remainder theorem. Remainder theorem is the same thing as the factor theorem. Factor theorem is just a special case of the remainder theorem, right? So this is called the remainder theorem. Remainder theorem. Factor theorem says this for a given x value, if f of x is equal to 0 then that x value is an x-intercept because y is equal to 0, right? This is called the factor theorem. Which basically says for a given x value, let's call the x value what are we going to call it? We don't want to call it x, let's call it w. For a given value for x if you sub in w here then if this is equal to 0 then you're on the x-axis, right? That's what we're trying to find here. Now the way we can do this is use synthetic division or straight out sub in 1, negative 1, 2, negative 2. So let's try to figure out what f of 1 is, right? f of 1 which means x is equal to 1, right? Which would make it x-1 as being a factor, right? So all you do you sub in 1 for x negative 6 times 1 cubed minus 1 squared plus 19 times 1 minus 6. This is equal to 1 cubed is 1 times negative 6 is negative 6. 1 squared is 1 19 times 1 is 19 minus 6. So what is this equal to? negative 6 and negative 6 is negative 12. negative 1 is negative 13 plus 19 is you're at 6. So what you just found was f of 1 is equal to 6. So you just found a point on the graph. When x is 1 y is 6, okay? When x is 1 y is 6. 1, 2, 3, 4, 5, 6. You just found a point on the graph. It's not the point we're looking for. So we're not going to use 1, x is equal to 1 in the synthetic division because we know it's not going to work. Let's try negative 1. So let's erase these guys. Let's plug in negative 1. Negative 1, negative 1, negative 1, negative 1. Negative 1 cubed is negative 1. Negative 1 times negative 6 is 6. Negative 1 squared is 1. So this becomes minus 1. Negative 1 times 19 is negative 19. Minus 6. So what do we got? 6 and negative 6 kill each other. Negative 1 minus 19 is negative 20. So f of negative 1 is negative 20. So when we go to negative 1, we're at negative 20. We're like way down here. 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20. Negative 1 and 20. So our graph goes from here to here crossing the y-axis at some point. And we know where it crosses it. It crosses it right there. Now we don't know if this thing is going up, touching this, coming back, going like this or anything. Right? So we can't just draw a straight line or a curved line. But we know from this that negative 1 is part of this. So this point was negative 1 and negative 20. This point was 1 and 6. We know that much. So whenever you're doing this work, you're not really wasting your time if you haven't found a factor right away. You're just finding points on the graph. Negative 1 and negative 20. So that didn't work. Let's use x is equal to 2. And this would have been x is equal to negative 1 would have been x plus 1 as a factor. Right? So let's try x is equal to negative 2. I'm going to erase it and rewrite it. Right? x is equal to negative 2. Or x is equal to 2. Sorry, not negative 2. Because I usually do positive and then negative. So what are we doing? Oh yeah, we're up there. 2 cubed minus 2 squared plus 19 times 2 minus 6. We can take this off too. We don't need this. 2 cubed is 8. 2 times 2 is 4. Times 2 is 8. 8 times negative 6 is negative 48. Negative 48. 2 squared is 4. So negative 4. 2 times 19 is 38 plus 38 minus 6. Now, this isn't going to equal 0. Because we're looking for 0. But let's figure out what it is anyway. So f of 2 is going to be negative 48 minus 4 minus 6 is negative 58 plus 38 is negative 20. Wow. 2 is negative 20. We're like over here again. So 1, 2. We're down to negative 20 again. I hope I did that, right? So 2 and negative 20. What a fluke. So check this out. We know that there's an x-sensor between this and that. And we know there's an x-sensor between here and here. Because it goes up and then comes down again. Which is pretty cool. So we got a constraint on the x-intercept. Interesting. Interesting. But it wasn't an x-intercept. So we know x is equal to 2 which means x minus 2 is not a factor of this thing. Let's try x is equal to negative 2. It could be pretty tedious. You do a lot of this. Good mental math. Good exercise. Hopefully I'll not make any mistakes. If I make any mistakes please let me know. Negative 2. Which means x plus 2. We're looking to see if x plus 2 is a possible factor of this. So this would be negative 6 times negative 2 cubed plus negative 2 squared plus plus or minus minus that would have been a bad thing. Plus 19 times negative 2 minus 6. Negative 2 cubed is negative 8. Starsky how you doing? Doing good doing math. Negative 2 cubed is negative 8 times negative 6 is 48. So 48. Negative 2 squared is 4 minus so it becomes minus 4. Negative 2 times 19 is negative 38. Minus 6. Oh snap crackled pop. Right? Negative 4 and negative 6 is negative 10. Minus 38 is negative 48. Plus 48 is zero snap. So f of negative 2 is equal to zero. That means when x is negative 2 y is zero. We found an x-intercept. Sweet. Negative 2 and zero. That's that point right there. That's what we needed. Because as soon as we find one of the x-intercepts we can do the synthetic division. It'll kick this down from degree 3 to a degree 2 and then we can use the quadratic formula to find the other two x-intercepts. Okay? So now that we did this let's do our synthetic division. The way we do our synthetic division is this. We're checking to see x is equal to negative 2. Negative 2. Which means we're looking to see if x plus 2 is a factor. Right? So our synthetic division is we bring this guy down. Negative 6. Whatever comes down here multiplies by that and goes up. So multiply negative 6 by negative 2 times negative 2 you get 12. And then you add those guys and you do the same thing. Negative 1 plus 12 is 11. Multiply by negative 2 is negative 22. Add them up. Negative 22 is 6 and you get 0. Right? Now I'm going to erase this part again. The remainder theorem and the factor theorem. So do you see what the difference between the remainder theorem and the factor theorem is? Everything else we did finding that point, that point, that point is categorized under the remainder theorem because the remainder was your y when you subbed in a specific x. When you find the y is equal to 0 for a certain x they say that's a factor theorem. I think that's stupid categorizing those two different things. I think what they should do is just eliminate the factor theorem and just call everything the remainder theorem and just put a caveat and they're saying if the remainder is 0 then you found the factor. That's what the factor theorem is. But they do what they do. They compartmentalize education of mathematics which is unfortunate. And remember, you don't need to be in a rush to copy all this down. You can just take a screencap of it. But if you want to take notes on the side for sure copy it down and take notes on the side. Now I'm going to erase this part and if I had a gigantic board we wouldn't erase this. By the way gang, thank you for the follows. Thank you for the subs. Thank you for paying attention. I hope you're liking this. And you're finding it useful. So we just found an x-intercept. Well what are these numbers? These guys check this out. What are these guys? These guys are the coefficients and the constant for when we divide this by x plus 2. So these guys actually are negative 6x squared plus 11x minus 3. It's what's left over when you take negative 6x cubed minus x squared plus 19x minus 6 and divide it by x plus 2. This is what you get. Writing it down as a division statement you simply have to appreciate whiteboards. You simply have to appreciate whiteboards. So what this means is this. Check this out. This I'm going to write it as a division statement. And then I'm going to erase it because we need the space down here. Negative 6 x cubed minus x squared plus 19x minus 6. If you divide this by x plus 2 you get negative 6x squared plus 11x minus 3. That's what it means. On the same note, if you cross multiply this guy up what we've done is this so far. Negative 6x cubed minus x squared plus 19x minus 6 is equal to x plus 2 times negative 6x squared plus 11x minus 3. That's what we've done so far. We broke this down into two things multiplied together. Now this is a quadratic. It's made up of two other polynomials of degree 1 multiplied together. So we need to factor this guy. How are we going to factor this guy? How are we going to factor that guy using the 4-step method I've shown you guys? The 4-step method is this. Should we do it here? I'm going to erase this whole thing and we're going to do the 4-step method and rewrite this thing. Watch this. We want to factor this guy. We'll use the quadratic formula. x is equal to negative b plus or minus squared of b squared minus 4ac over 2a. Before we use the quadratic formula, I'd like to see if we could factor this manually. We're looking for the x-intercepts because this is a new function. You can consider it to be a new function. This is really f of x times x plus 2 which is one function. It's called as g of x. You could say let g of x equal x plus 2 and let k of x equal negative 6x squared plus 11x minus 3. So f of x is really g of x times l of x. Let me write this down properly. k of x times k of x where g of x was x plus 2 and k of x is this guy. So we can say this is k of x. So we want to find out when this is equal to 0 which means when it crosses the x-axis. Hey, Chico, happy anniversary. Thank you, Eduardo. Now, let's use that four-step method that I've shown you guys to factor. In my part of world, they teach people decomposition. Decomposition sucks. I use the four-step method and I wrote it down as four-step method and this is factoring complex trinomials. Factoring complex trinomials. If you look for factor and complex trinomials online and put a Chico in the search, there's going to be a bunch of videos that pop up. So this is the way you factor complex trinomials. Negative 6x squared plus 11x minus 3 is equal to 0. Take this negative 6 multiplied by this. You get x squared plus 11x negative 6 times negative 3 is 18. Oh, no, no, hold on. Before we do this, we've got to factor out a negative 1. My apologies. Factor out a negative 1. We don't want the first term to be negative when we're doing this, right? So factor out a negative 1. Jump in the gun. Then we're going to get 6x squared minus 11x plus 3 is equal to 0. Now, don't divide the negative 1 out. You need the negative 1. Then you take the 6 and multiply by this. So you've got negative 1. Then you're going to get x squared minus 11x 6 times 3 is 18 is equal to 0. Now, you're looking for two numbers that multiply to give you 18 out to give you negative 11. Do you know what they are? Because they do exist. xx Two numbers that multiply to give you positive 18 and out to give you negative 11. This is what I tell people. It's really ridiculously important to to make sure you know your multiplication table. Turning slow mode off. Cheech will disable slow mode for this row. No slow mode. I'm going to give you the answer. I'm going to give you the answer. I'm going to give you the answer. Negative 2 and negative 9. Minus 2 minus 9. Negative 2 times negative 9 is 18. Negative 2 plus negative 9 is negative 11. So we did that. But this is a factor of this. It's not a factor of that guy. The way you use this method is you take the 6 drop it in the front of the x's and then you take out the GCF and dump. So you do GCF so you got negative 1. What's the GCF from 6x minus 2? Well 2. You can take a 2 out from 6 and a 2 out from 2. So you take the 2 out and dump it. So now you got 3x minus 1. What's the GCF out of 6 and negative 9? Or 6 and 9 is 3. Take the 3 and dump it. Now you got 2x minus 3 is equal to 0. What does that say? Now before we finish this we're going to multiply the negative 1 either into this guy or into that guy. It doesn't make a difference. It shouldn't make a difference. Because that way we get the real factor of it. I'm going to multiply this into here. So the factors of this are negative 3x plus 1 and 2x minus 3. Now remember you only have one negative 1 here. So you can only multiply the negative 1 into one of these. Do not multiply them into both. So whenever you have a negative quadratic leading coefficient factor out a negative 1 and then use this method and then multiply the negative 1 back in. What this tells us is the other 2x intercepts are we've talked about this. You can set each one of these equal to 0. So you can go negative 3x plus 1 is equal to 0 which means 3x is equal to or negative 3x is equal to negative 1. So x is equal to 1 over 3. That's the other x-intercept. And that basically means this guy crosses the y-axis here and goes through a third and then goes up to here. And the other x-intercept is 2x minus 3. 2x minus 3 is equal to 0 so x is equal to 3 over 2 which is 1.5 which is exactly what we got. So we knew this guy came up went through here, went through that and we got our 2x intercepts right there. Cool? So I'm going to erase all this and I'm going to write this guy in its factored form. That way we see exactly where the x-intercepts are. What I'm going to do is I'm going to extend this this way so we can zoom in into the location where things are occurring. So let's take, get rid of this. What do we got? I'm going to write this up here so we don't use it. Negative 3x plus 1 and 2x minus 3. Now obviously when you're doing a work you would never erase this stuff. And we didn't have to use the quadratic formula. Quadratic formula. Cool? And if we did this, this is what we ended up getting. Negative 3x plus 1 and 2x minus 3. So what we have now our original function f of x right? How many swiggles is that? How many sw- is 3? Matt? So what we got is f of x which is 6x cubed minus x squared plus 19x minus 6 can be factored into x plus 2 times negative 3x plus 1 times 2x minus 3. Right? Now because these things are so tight here, I'm going to rewrite these things and I'm going to make my axis stretch out a little bit so we can zoom in on it. So the three points, four points we have right now are these. Negative 2 and 0, which we already know negative 1 and negative 20 1 and 6 and 2 and negative 20. I'm just writing them down here because I'm going to kill them and I'm going to put them back on so we don't forget. And there's nothing wrong with adjusting your graph once you figure out what a function looks like, right? So what I'm going to do is I'm just going to draw my line again and I'm going to make this one and this 2 and this negative 1 and this negative 2. Just stretching it out. These are the points we're going to put on here. Negative 2 and 0 that was this point right there. That's going to be negative 1 and negative 20, negative 1 and negative 20. And then we got 1 and 6 and then we got 2 and negative 20. So this point is 2 and negative 20. This point was negative 1 and negative 20. This point was 1 and 6 and that point is negative 2 and 0. Let's erase these guys. Then we got so this is negative 2 and 0. So our X intercepts based on this because we're going to set this equal to 0 are going to be X is equal to negative 2 which we got. X is equal to 1 over 3 and X is equal to 3 over 2. So 1 over 3 is here and 3 over 2 is just 1 and a half which is here. Right? So this is 3 over 2 and 0. This is where we're going to put this. A third and 0. Right? So right now we know graph does this. It's a negative coefficient in front of the leading coefficient so we know it goes like this. This thing is going to come down go down here somewhere most likely come up or go down here and then hit it on the way up and the peak we don't know where it is. Now think about it this way. I'm going to put this let's do a green red. Let's put it on red. Hopefully the red will show up. Will the red show up or the black contrast? Let's see. Is there a contrast? Really? Let's see if the red is a contrast. Is that a contrast? Green? Let's see if the green is a contrast. Yeah, the green is a contrast. Right? 10x-3y equals 18 solving system solve system by elimination. We can do system by elimination after this. I'll take a look at this. So the green. So we know the graph goes through this. We know the graph goes through this but we don't know if it comes down this way or if it goes goes down here and then comes back up and then we don't know how low it goes. We know it goes through this goes through this but we don't know if it goes through this and goes peaks up there or peaks up here so let's assume it can go up there, come down, go through this so we don't know the feel of it exactly how it looks so what we need to figure out right now is this point and this point. Those two points we get from the derivative because the derivative is the slope of the graph of the original function. So this is a function that gives you the slope of this function at every point on x. So if you want to find out what the slope of the graph is when x is equal to negative 2, you plug in negative 2 here, find it and that's the slope, rise over run. Here we can do one. Let's do it here in green. So let's assume we want to find out what the slope of the function is of this original function is at x is negative 2. So we can do this, find f of negative 2 which is going to be negative 18 times negative 2 squared minus 2 times negative 2 plus 19. Negative 2 squared is 4 times 18 is 4 times 20 is 80 less 8 so 72 should be 72 4 or 72. So negative 72 negative 2 times negative 2 is 4 plus 19. So that's going to be 23. What's 72 minus 23 6, 12, 9, 4 so it's negative 49. What does that mean? Negative 49, that's the slope. I already heard that from Mike Oxlok. What's negative 49 mean? That's rise over run. Slope which is rise over run and it's negative so the slope is going like this, really sharp. So for every one unit in the x direction you're going down 49 units. Rise over run. That's what the slope is. That's what this gives you. So for one unit going this way, we've gone all the way down to 49. Negative 49. So if we plug in negative 49 should it be negative 49? Did I do that right? 4 squared, 1, 19, 18, 72 looks like it but it's weird because it should be a negative 20 unless we did something else wrong previously. So there's a little bit of discrepancy in what we did here in finding where is at that point. But the function is correct. Is it? Is it? Yes it is. 3, 18, 2, 2, 2. Yeah, so that's correct. That's what I'm checking right now. Your rise which is up and down is y and your run is x left and right. Indeed. Rise over run. The y is up top. So this is the y and this is the x. So this means negative 39 over 1. That's how much you're going in the vertical 1 in the horizontal. Gay bitches for it. I don't know. So check this out. If this graph or if this function this guy here the derivative of this is the slope of our function. Right? What we have to realize is what you have to avoid. What you have to realize is at the points of turning whoop whoop where it turns the slope is zero. He's showing us right. He explained better than my teacher Lazarus. So what happens is check this out. Let's assume we're zooming into here. Right? Let's assume the graph is going like this and then turns around and comes down. And we're looking for this point. Right? Let's assume this is this part which is zoomed up here. Right? So if that's the graph then the slope of this function is like this. Goes up, goes up, goes up, goes up, goes up. The slope is doing this. The slope is doing this. The slope is doing this. The slope is but it has to turn around and come down again. Right? Because this way is positive, positive, positive, positive. At this point the slope of the graph is going to be zero. Right? Like literally zero. Right? And then the slope starts coming down again. Starts going negative. So if we set this function which is the derivative of this function giving us the slope of this function if we set this equal to zero solve for x we're finding the x values where this thing turns around. We're finding that x value. Cool. And then once we found that x value we can plug it back into the original function and find the y associated with it. Right? Has to be zero. So let's do. What am I going to erase? We need space. I'm going to take all of this down and what I'm going to do is I'm going to, hold on, let me erase this. You can see that you're doing a lot of work. Right? I'm going to write the factors of this function up here. That way we have it. Right? So this guy is really x plus 2 negative 3x plus 1 2x minus 3. Okay? That was our purpose. We want to find the x-intercepts. Right? So I'm going to erase all of this all the way up to here. That way we got room to play. Right? Do we need anything else from here? Nope. Yes. All that gone. Isn't that nice? Let's build it up again. I hope you're having nice fluids. So take a look at this. Should we do this in green? Nah. Let's stick with our purple. We'll use the green. Ellipsoid nose cone, my friend. Ellipsoid nose cone. I don't know what that means, but sounds cool. Faster than sound. Now watch this. We want to find when f prime of x negative 18 x-squared minus 2x plus 19 is equal to 0. We could take this and multiply by this and try to find two numbers and multiply it to give you whatever that multiplication is and ask to give you negative 2 but I'm not going to mess around with that. I'm going to use the quadratic formula. It might be ugly, but it's going to give it to us directly. So x is equal to negative b plus or minus squared of b squared minus 4ac over 2a which is going to be and don't forget gang. Free assange, free assange, free assange, right? So are a here write it like this. a is equal to negative 18 b is equal to negative 2 and c is equal to 19 it's just a coefficients and the constant. Right? So let's plug it in. x is equal to negative negative negative 2 is 2 plus or minus squared of negative 2 squared minus 4 times a times c which is 19 all over 2 times negative 18 which is going to be 2 plus or minus squared of negative and the negative is positive now we got to go 4 times 18 times 19. What is that? 4 times 18 times 19 4 times 18 19 1368 1368 plus 1368 all over 2 times negative 18 is negative 36 now what are the odds of 4 plus static being a perfect square? I doubt it. Plus 4 and then we take the square with well it's not obviously. Better and so do mistakes so do mistakes so oh what was that number? We wanted that number so this one this ends up being 2 plus or minus square root of 1372 1372 over negative 36 which means this is equal to 2 plus or minus 37 I'm just going to go with the decimal 0.0405 0.0405 I'm just rounding, right? Welcome in the stream that was nice over negative 36 so X these two points occur and X is equal to 2 plus 37.0405 over negative 36 and X is equal to 2 minus 37.0405 over negative 36 now we have to figure out what this is so plus 2 plus 2 and then divided by 36 divided by 36 but negative so it's going to be negative negative 1.084 and this one I'm going to explain to you what that is that value right there is this negative 1.8 so where I drew this is incorrect this graph actually comes down here and the point that it turns up again is close to this one but it's on this side okay it's on this side so it goes like this and then goes up this one is going to be negative it's going to be 2 minus 37.045 whatever so it's going to be negative 35.0 405 divided by 36 which is 0 really close 0.9 because it's negative over negative it's positive 0.9733 so where does that occur that occurs right before the one so that one we had at the right location so what we have right now is this let me erase these so you see what we've got crazy bro Athens how are you doing hi everyone hope you're all doing fine you too you too so what we have here is this part was incorrect because where's our green the x part of this is this so I'm going to erase this that was 1 remember right so that's 1 and 6 let me write that down here this is 1 and 6 this point here is 0.9733 and we need to find a Y we need to find Y how are we going to find Y we just plug this into this and we find our Y cool right it's been a while since I caught a live stream the curious K long time no see chichou keep mathin it up keep mathin it up bro that's what the plan is and then this one is this let's bring it down comes here really close and then zooms back up right and this point here is going to be negative 1 negative 1.08 45 and the Y and we need the Y for that one too and the Y for that one again we plug it into the original X and we find it so let's do it let's see what we got I'm going to erase all of these right now but what I'm going to do is I'm going to write down the X intercepts or not the X well they are the X intercepts really but negative 1 let's leave us some room negative 1.08 45 and 0.9733 733 and we need to find the Y associated with this so now I'm going to erase all this and give us more space right so let's kill all this and I'm going to erase these guys too we don't need that just makes it too much and I'm going to kill this right and I'm going to kill this clean clean sweet all these equations represent or just figures in this case lines in this case lines thank you for the follow-up so now what we need to find it's just not a Y it's F of this point and F of this point in the original function so let's do the first one F of negative 1.0845 is equal to now you can plug this in either in the expanded version or the factored version the factored version is easier to plug in I think in my opinion right so you could plug into this negative 1.0845 plus 2 times 3 negative 3 times negative 1.0845 plus 1 times 2 times negative 1.0845 minus 3 instead of doing cubes and squares and stuff like this right so I need to do that with the calculator but what I'm going to do with that or if someone wants to do that if you let us know what it is confirm what I'm doing because I'm going to do it with Anna what do you call it on the computer so sometimes I make booboo's 0, 8, 4 5 negative plus you need to use the calculator I need to use the calculator no chicho I know it so this one is the numbers aren't nice and neat and that's one thing I do again when I'm working with students I make the first level of numbers relatively easy to do but then the rest of it is what it is right so they're not nice and clean they come out to be real life numbers so this times this is 3 or times that is going to be positive and then I'm going to add 1 so it's going to be 4. 2535 oops and then it's going to be 2 times 1 1.0 8 4 5 oh I forgot the 1 scooper 2 times she shows birthday today 1.0 8 4 so it's going to be negative and then mine is that so it's just going to be plus 3 it's just going to be 5 point 169 but negative right and this one positive yeah so for this when it's this we've got 1 negative so it's a negative times a positive times a positive so we know it's negative I can only do linear algebra and tell you if a function a figure is a function of another cool calculators are fine calculators are fine really so now we've got to multiply all these guys together so what does this guy is equal to let me bring it up again so that's going to be times 4.2535 4 point 2535 times 0.9155 0.9 1 5 which is 20.12 8 5 half of negative 1.0845 is that which is that point right there so that point right there where's our green this point right there is 20 negative that should be negative negative because it was a negative negative 20 point 1 285 so that's our first minimum it's going to be relative minimum because over here it goes down further so negative 1.0845 and negative 20.1285 that's the bottom now we're going to do it for this guy so this guy is negative 20.1285 let's do it for that one half of 0.9733 is going to be 0.9733 plus 2 negative 3 times 0.9733 plus 1 2 times 0.9733 minus 3 0.9733 minus 3 let's do it again this one is easy this one is 2.9733 times 3 times that at 1 did I do it right yeah so let's see what we got 0.9733 0.9733 times 3 which is negative so we add 1 so it's going to be negative 2.9199 negative 2 what 0.9199 right and then 2 times 0.9733 is equal to that make it negative at 3.1 it's going to be negative so it's going to be negative 1.0534 and then we're going to multiply all these together right times 2.9199 0.9199 times 2.9 2.9733 which is 9.1453 so f of this is equal to that so the other point is 0.9733 and 9.1453 this point becomes which we had a pretty good spot but it should be further down if that's 6, 7, 8, 9 so it should be further down here it goes like this that point is going to be 9.1453 cool and don't forget game freesage freesage what really got my noodle going was learning how old this is with the exception of functions of course so we found that, that now what we need to find is that where this guy is equal to 0 I believe we're going to set it equal to 0 where it changes am I correct? should be because there's inflection points what happens at a certain point it goes like this we want to find this point and actually what we do for that is find where it's positive and where it's negative because positive means it's concave up and negative would mean it's concave down that's what it really gives us right? but what we need to do is find where it's 0 because that's the break point where it goes from positive to negative okay so let's do the second derivative I'm going to erase all this by the way again let me put this point up here too 9 points 1, 4, 5, 3 those are are relative minimum and relative maximums this and this this guy is relative x that point and min relative min which is that point okay now let me erase all this the last thing we need to do is find the inflection points where it's switching up so all we're going to do is find where it's equal to 0 I believe so anyway so set oops, that's not a s that's an f set f double prime of x is equal to 0 so negative 36x minus 2 is equal to 0 so negative 36x is equal to 2 divided by negative 36 so x is equal to negative 1 over 18 negative 1 over 18 so right about here right? if you break this from that's negative 1 negative 1 over 18 is like right here right? really low so here comes down goes up so we can solidify this now because we know what it looks like right? comes up and at a certain point here it turns around again and goes down oops my graph sucks at this point right? this point right there which is negative 1 over 18 and we need to find the y associated with that so we're going to plug in negative 1 over 18 for x and find what the y is because you're always going back to the original function right? this is by the way seriously distorted perspective my graph sucks okay so we're going to find f of the original function of negative 1 over 18 how we're going to do that let's do it in that as well so that's going to be negative 1 over 18 plus 2 times negative 3 negative 1 over 18 plus 1 times 2 negative 1 over 18 plus 3 I'm just going to simplify this just do algebra so common denominator here is 18 that's negative 1 that becomes plus 36 3 goes into 18 6 times negative and negative is positive common denominator 6 so that's 1 plus 6 2 goes into 18 9 times common denominator is 9 9 times 3 is 27 so you've got negative oops that's a 9 9 1 minus 27 right? destructive graphs can be called, graphs can be called original arts in some ways crazy broad things who asked this question? who asked this question? I forget who it was that asked this question Felix? I think Felix asked this question so we're going to multiply all this crap together look at all these numbers 35 I sort of came up with the numbers the numbers are horrendous but they are what they are I'm not going to sugar or what do you call it make it if I picked it randomly maybe it would have been easier or try to thought about the numbers a little bit better so we need to multiply this out nothing really simplifies we've got 2 that goes into both of these but that's about it well 2 goes into this 3 times 2 goes into that 19 times negative 19 that's it so we've got to multiply all this out I picked crazy numbers so let's check this out this is a lowest common denominator no we're multiplying we're not adding so we don't need the lowest common denominator when you're multiplying you're just reduced before you multiply unfortunately this is all it does it doesn't reduce anymore so it's going to be 35 35 times 7 times 19 which is 4,655 4,655 and it's negative over 18 times 3 times 9 3 times 9 is 27 so 27 times 18 equals 486 so it's going to be 4,655 divided by 486 4,655 divided by 486 486 you get negative 9. 578 2 that's there which is not bad the graph should be a little bit lower so on this side this is concave up it would be positive this guy it's above 0 and on that side it's negative tomorrow I have my final exam for the semester in econ I'm a bit stressed so this point this guy is now negative 9.578 2 and we found all the important points of this graph so we've basically defined this function we got everything we need for this we found all the critical numbers we found this this this and this that's it I'm glad we didn't do a 4th degree 4 I hope that was clear enough that's sort of intro intro to calculus best of luck indeed fan fan James Bond thanks a lot don't get stressed don't get stressed don't get stressed Lazaro 5 out of points fun I'm glad we've gotten into calculus I still haven't created any calculus lessons straight up like edited video but we will at some point once we get back into making edited videos and stuff definitely an ASMR format definitely an ASMR format fun easy peasy I have something that I didn't understand can you try to solve it? sure let's try so I came here later why did you find the point we need to find where the curve switches right? because it's the inflection point for example a cubic function does this we found this point for this one it was the other way around right? so we're finding this point we're finding this point okay oh yeah this one solving system by elimination okay got it cool so you have two functions so system by elimination and stuff do you understand what it is you're trying to do if you're solving a system by elimination you're trying to find out where they cross 10x-3 is equal to 18 10x-3 is equal to 18 and minus 3y minus 3y is equal to 18 and negative 2x plus 3y is equal to 6 negative 2x plus 3y what? 3y again cool 3y is equal to 6 and you want to solve by elimination? sure this is your first equation this is your second equation you want to find out where these two functions cross each other 10x-3 oh that's all German French funny so check this out in English I'll do it so you're trying to find out when this function crosses this function you can rewrite both of these equations in terms of y is equal to mx plus b and graph them and see where they cross if they cross I'm not going to bother doing that because you just want to solve it by elimination solving it by elimination means get rid of one of the variables right off the bat for us if we just add the two functions the y's will disappear because that's 3y that's negative 3y so they kill each other we could also multiply a second equation by 5 this will be negative 10x plus 15y is equal to 30 and then add them up and kill the x's but I'm not going to do the extra stuff I'm going to kill the y's right off the bat so if we add these two equations this kills this this becomes 8x is equal to 24 divide by 8 so x is equal to 3 so we found the x you can find a y by subbing in the x into this one and this one I'm solving for y or you can do elimination again you can take equation 1 and go okay let's keep that 10x minus 3y is equal to 18 for equation 2 multiply it by 5 if you multiply it by 5 you're going to get negative 10x plus 15y is equal to 30 now add them if you add them these guys kill each other this becomes 12y is equal to 48 divide by 12 so y is equal to 4 so where they cross is solution is 3 and 4 in xy plan u have to give the equation line that is passing through this so give the equation of a line I've already done that but you need a slope as well we don't have a slope what's the slope fan I hope that's thank you my pleasure Lazaro so we have two points I think if I'm reading this correctly wait a second line that is passing through in xy plan u but when you write down 1 2.3 is that an x-intercept y-intercept or is that a point or is your point this is that the point where this is your x and that's your y sorry 2 3 is that what you got yeah that's an equation of a line